To complete the square for the quadratic equation \( x^2 + 8x = 9 \), follow these steps:
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Start with the equation: \[ x^2 + 8x - 9 = 0 \]
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Move the constant to the other side: \[ x^2 + 8x = 9 \]
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To complete the square, take the coefficient of \( x \) (which is 8), divide it by 2, and square it. The calculation is: \[ \left(\frac{8}{2}\right)^2 = 4^2 = 16 \]
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Add and subtract this value (16) to the left side of the equation: \[ x^2 + 8x + 16 - 16 = 9 \]
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Simplifying gives: \[ (x + 4)^2 - 16 = 9 \]
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Now, move the \(-16\) to the right side: \[ (x + 4)^2 = 9 + 16 \] \[ (x + 4)^2 = 25 \]
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Take the square root of both sides: \[ x + 4 = \pm 5 \]
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Solve for \( x \):
- For \( x + 4 = 5 \): \[ x = 5 - 4 = 1 \]
- For \( x + 4 = -5 \): \[ x = -5 - 4 = -9 \]
Thus, the two values that solve the equation \( x^2 + 8x = 9 \) are: \[ \boxed{1} \text{ and } \boxed{-9} \]
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